Value at Risk vs Expected Shortfall Explained
Value at risk (VaR) is a loss threshold associated with a specified confidence level and holding period under a statistical model. Expected shortfall (ES) estimates the average loss in the tail beyond the corresponding threshold, under the defined model. Neither statistic is a guaranteed worst-case loss, and both can be misleading if the model or historical sample misses extreme events.
What a VaR number means
A one-day 95% VaR of £10,000 conventionally means that the model estimates losses greater than £10,000 on roughly 5% of days under the assumed distribution. It does not say that the maximum possible daily loss is £10,000, or that losses beyond it are capped. Exact quantile definitions and treatment of discrete observations can vary. Always specify probability level, holding period, portfolio and whether the number comes from historical, parametric or simulated data.
Expected shortfall focuses on the tail
Expected shortfall estimates the average loss conditional on outcomes in the worst specified tail, or an equivalent tail-average definition. With the same 95% confidence level, it looks at the worst approximately 5% of outcomes. It can therefore distinguish portfolios with similar VaR thresholds but radically different extreme losses. ES is still an estimate and can be unstable when observations of genuinely extreme scenarios are scarce.
Worked example using ten losses
Imagine ten equally likely hypothetical one-day outcomes, expressed as losses in pounds: 0, 100, 200, 300, 400, 500, 600, 700, 1,000 and 5,000. For illustration, the nearest-rank 80% loss quantile is the eighth ordered observation, £700. The worst 20% consists of £1,000 and £5,000, giving an average tail loss of £3,000. Different statistical interpolation definitions may produce a different estimated VaR. This deliberately simple example shows why tail severity matters.
Why VaR can hide extreme outcomes
Two portfolios can have similar normal daily variation and therefore similar VaR estimates but different exposures to rare jumps or illiquidity. A short-option strategy may collect small frequent premiums while facing occasional very large losses. Ordinary return distributions can exhibit skewness, fat tails and clustering that violate a simple normal model. A precise-looking VaR output is not a substitute for testing severe market scenarios.
Historical versus parametric methods
Historical VaR uses observed portfolio or market changes from a selected sample, while parametric methods assume a distribution and estimate its parameters. Monte Carlo simulation draws outcomes from a model. Each can fail when the future differs from its assumptions. Historical windows can miss unfamiliar crises; parametric models can understate tail risk; simulation results are only as sound as the process generating them.
Liquidity, horizon and scaling
A one-day risk estimate cannot simply be converted to a ten-day figure using square-root-of-time scaling under arbitrary conditions. Asset returns can show volatility clustering, serial dependence and jumps. A portfolio that cannot be liquidated within the assumed horizon also faces market impact. For large positions, stress liquidity and price gaps separately from statistical loss quantiles.
Basel context versus personal trade rules
Regulatory bank market-risk frameworks use expected shortfall within broader capital and stress-testing methodologies. That institutional framework is not a universal rule for an individual’s position sizing. Portfolio-specific financial risks include leverage, counterparty failure, basis and concentration that a single estimated ES may not capture. Stress testing and governance remain important.
Risk-reporting checklist
State model, data sample, horizon, confidence level, loss sign convention, currency and portfolio composition. Report VaR and ES alongside backtesting exceptions, severe scenarios and liquidity assumptions. Explain how conclusions change with different windows or stressed regimes. Never use VaR as a promised maximum loss or a substitute for understanding the underlying instruments.
For additional risk and market-structure context, refer to portfolio drawdown and risk measurement.
Worked example: 95% VaR and expected shortfall
The following original example uses hypothetical inputs to illustrate the method; it is not an actual market quotation.
| Statistic | Hypothetical loss distribution | Result |
|---|---|---|
| Scenario set | 100 equally weighted one-day outcomes | 5 worst losses form the 5% tail |
| 95% VaR threshold | Assumed cut-off loss | £4,000 |
| Worst five losses | £5k, £6k, £7k, £9k, £13k | Total £40,000 |
| 95% expected shortfall | £40,000 ÷ 5 | £8,000 average tail loss |
| Important distinction | VaR is a threshold; ES averages severe outcomes | Neither is a maximum possible loss |
This deliberately constructed scenario distribution has the worst five losses all above the £4,000 threshold. At a 95% confidence level, VaR reports a loss threshold associated with the estimated tail frequency, while expected shortfall describes the average loss beyond it. Different statistical quantile conventions, scenario weighting, horizons and models can produce different numbers. Estimates also depend on data quality and the treatment of rare events; even a calculated £8,000 expected shortfall does not cap losses at £8,000.
Frequently asked questions
Does a 95% VaR cap losses?
No. The model explicitly allows worse outcomes in the remaining tail.
Why is expected shortfall different?
It measures average severity beyond a chosen loss threshold rather than only the threshold.
Are ES estimates always reliable?
No. They depend on model choices and scarce tail observations.
Sources and further learning
Continue with Trading Strategies Explained: Rules, Edge and Risk.